No abstract

All over the globe, soft set theory is a topic of interest for many authors working in diverse areas because of its rich potential for applications in several directions since the day it was defined by Molodtsov in 1999. Moreover, soft set theory is free from the difficulties where as other existing methods viz. probability theory, fuzzy set theory. Considering to these benefits, soft set theory has became very popular research area for many researchers. To contribute this research area, in this paper, we examine some properties on soft topological spaces such as neighborhood structure of a soft element and soft interior, soft closure, and soft cluster element and so on that are based on soft element definition that gives us a different perspective for development of soft set theory. Moreover, we give some examples to clarify our definitions.

In this paper, the concept of soft interval is given and an example for soft Scott topology is illustrated by using the soft intervals. A tabular form for all soft closed intervals is presented. Then soft order topology is introduced and some application of it are expressed. Also we show that, the Soft Scott Topology and Soft Order Topology do not have to be same even on the same soft set.

Molodtsov introduced a theory of soft sets which can be seen as an effective tool to deal with uncertainties in [16] and then, the development of the theory gradually increased. In this paper we define the notion of soft semi-topological groups based on the notion of soft element which is defined by Wardovski in [28]. Also some theoretical results are given. Thus soft topological group theory become open to improvement with these new definitions and results.

In this paper the tolerance soft set relation on a soft set is defined and some examples are given with their matrix representations. Also, pre-class and tolerance class concepts for a given tolerance soft set relation are introduced and some examples related to these definitions are illustrated. Some theoretical results are proved such as every pre-class contained by a tolerance class and intersection of two pre-classes is a pre-class as well. Moreover, a method to find out the tolerance classes and pre-classes by using matrix representation of a tolerance soft set relation is explained with examples.